#94817

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	C₂²⋊C₄
commutator	C₂²	C₂×C₆₄
Frattini, cosocle	C₂×C₆₄	C₂²
socle	C₂	C₂²⋊C₆₄
prime radical	C₂³	C₆₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₂×C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

53 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,3,10,4,11,16,35,5,12,17,36,20,39,49,75,6,13,18,37,21,40,50,76,24,42,52,78,55,81,91,110,7,14,19,38,22,41,51,77,25,43,53,79,56,82,92,111,27,44,54,80,58,83,93,112,61,84,94,113,95,114,119,126,8,9,15,30,23,31,45,65,26,32,46,66,57,69,85,100,28,33,47,67,59,70,86,101,62,72,88,103,96,106,115,121,29,34,48,68,60,71,87,102,63,73,89,104,97,107,116,122,64,74,90,105,98,108,117,123,99,109,118,124,120,125,127,128);(2,9)(3,15)(11,31)(12,32)(13,33)(14,34)(16,45)(17,46)(18,47)(19,48)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(81,106)(82,107)(83,108)(84,109)(91,115)(92,116)(93,117)(94,118)(114,125)(119,127)

Polycyclic presentation:

2^9|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ci,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2013);